House Republicans plan to bring up a vote to repeal the health-care overhaul early in the new Congress that opens Wednesday, at least before President Obama delivers his State of the Union address later this month, a key Republican lawmaker said Sunday.

Rep. Fred Upton (R-Mich.), incoming chairman of one of the House committees that oversees health policy, said undoing the Democrats' health reform law would be a top priority for the new GOP-controlled Congress.

Upton said on "Fox News Sunday" that he believes there may be enough opposition in the new House to reach the two-thirds majority required to override a presidential veto. Short of that, he said House leaders will "go after this bill piece by piece."

"As part of our pledge, we said that we would bring up a vote to repeal healthcare early," Upton said. "That will happen before the president's State of the Union address. We have 242 Republicans. There will be a significant number of Democrats, I think, that will join us. You will remember when that vote passed in the House last March, it only passed by seven votes."

Upton added: "I don't think we're going to be that far off from having the votes to actually override a veto."

"We will put forth a clean repeal bill of ObamaCare and you'll continue to see us make that fight, because that's what the American people want us to do," Rep. Michele Bachmann (R-Minn.) said on CBS's "Face the Nation."

But while Republicans may muscle through a repeal bill in the House, its prospects are slimmer in the Senate, where Democrats and independents will enjoy a 53-47 majority.

"If we pass this [repeal] bill with a sizable vote, and I think that we will, it will put enormous pressure on the Senate to do perhaps the same thing," Upton said. "But then, after that, we're going to go after this bill piece by piece."

Upton said Republicans would look to overturn key portions of the health-care law, including the individual mandate, a requirement for businesses to complete 1099 tax forms and the so-called Stupak amendment that allows federal money to be spent on abortions in cases of rape or incest.

"We will look at these individual pieces to see if we can't have the thing crumble," Upton said.

Democrats said they plan to aggressively defend Obama's legislative accomplishments, chief among them the health-care bill.

"They're talking about wasting time repealing health care, when they know that the Senate and administration won't go along with it. Don't waste time. Create jobs," Rep. Steve Israel (D-N.Y.), chairman of the Democratic Congressional Campaign Committee, said on CNN's "State of the Union."